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A square root function involves taking the square root of an expression, forming the basis of many algebraic functions. The function \( f(x) = \sqrt{x+2} \) is an example where the expression inside the square root is \( x+2 \). For any square root function, it's important to ensure that the expression inside the root is non-negative because you cannot take the square root of a negative number in the set of real numbers. This ensures the function outputs real number values only. Square root functions typically exhibit a curve that gradually increases or decreases. Observing the graph of a square root function helps in understanding its behavior as it progresses along the x-axis.

Inequalities are mathematical expressions that describe the relative size or order of two values or expressions. They use symbols such as \( > \), \( < \), \( \ge \) (greater than or equal to), and \( \le \) (less than or equal to). Solving inequalities involves finding the values that satisfy the inequality condition. In the exercise, we deal with \( x + 2 \ge 0 \) since the square root function requires the inside expression to be non-negative. Here, the symbol \( \ge \) implies that \( x + 2 \) can either be zero or any number greater than zero. To solve \( x + 2 \ge 0 \), we isolate \( x \), resulting in \( x \ge -2 \). This simple inequality solution tells us over what values the function is defined, illustrating the fundamental role inequalities play in defining function domains.

The domain of a function is a set of all possible values that the input \( (x) \) can take where the function still returns real number outputs. Understanding the domain is crucial as it defines the boundaries over which a function is valid or makes sense. For functions involving a square root, like \( f(x) = \sqrt{x+2} \), the domain is determined by the condition imposed on the inside expression of the square root. Here, the domain is \( x \ge -2 \). This means any \( x \) value that is equal to or greater than -2 is suitable for the function. Knowing the domain helps avoid errors in calculations and ensures that all inputs analyzed produce valid outputs. Understanding the domain allows for better manipulation of the function further either for graphing or applying it to real-world situations. Recognizing function domain is a foundational skill in algebra and calculus.